Reduction and Elimination in Philosophy and the Sciences

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Let T I be the minimal theory of truth with full induction: TI : = Q ∪ ( IndP ) ⋅ Lτ . Theorem 1 TI and every pure extension of T I is essentially reflexive. Corollary 1 TB , UTB, PT, TC are essentially reflexive. For other conservative theories with restricted induction it is far more complicated to show that they are reflexive. One example is Tarski's compositional theory with restricted r induction. In (Halbach 1999) the conservativity of TC over PA is proved by a cut elimination proof. This proof has at least two advantages in comparison to a model theoretic proof along the lines of (Kotlarski et al. 1981). First it can also be used for other base theories especially for all IΣ k r with k ∈ω . ( IΣ ) : = IΣ ∪ ( C1) − ( C8). TC k k Second it can be formalised in a way that makes it provable in PA . So we get: Theorem 2 For every k ∈ω : PA proves ∀ x( Sent( x) ∧ Pr r ( x) → Pr ( x)) TC ( IΣ ) IΣk k r With this it can be shown that TC proves the consistency of all of its finite subtheories. Theorem 3 r r PT , TC are reflexive. Proof: Theorem 2 shows that for any k ∈ω : PA proves ConIΣ → Con r . Since PA is reflexive, all IΣ k are k TC ( IΣk ) finitely axiomatizable and r TC proves Con r TC ( IΣk ) Interpretability Relations of Weak Theories of Truth — Martin Fischer r TC PA ⊆ , for every k ∈ω : , which is enough to show that r r TC is reflexive. A similar argument shows that PT is reflexive. □ Theorem 4 − − PT , TK are not reflexive. Corollary 2 r r PT , TC are not essentially reflexive. For extensions of PA reflexivity and relative interpretability, p , are connected by Π1 -conservativity in the following way as shown for example in (Lindström 1997): Theorem 5 Let PA ⊆ T . PA conservative over PA . This shows that: T p iff T is reflexive and 1 Π - Theorem 6 r r TB , UTB, PT , TC are relatively interpretable in PA . On the other hand it is easy to see that theories that are not reflexive or Π1 -conservative over PA are also not interpretable in it. Theorem 7 − − PT , TC , PT, TC are not relatively interpretable in PA . Relative interpretability in PA implies reflexivity and Π1 - conservativity over PA but it does not not imply conservativity over PA . Theorem 8 TB + ¬ Contb is relatively interpretable in PA but not conservative over PA . 4. Weak Theories of Truth Considering the set TT of all theories of truth, that is theories formulated in L τ and containing PA , there are the two subsets with one criterion of weakness: CTT : = { T ∈TT T is a conservative extension of PA } . ITT : = T ∈ TT T p PA } . { The combination of these two criteria allow a more fine grained picture of theories of truth, especially for weak theories. In the preceding sections it was shown that CTT, ITT , their complements and their combinations are nonempty. There are four possibilities of combination. Strong theories of truth not fulfilling either of both criteria will not be investigated here. The set ITT consists of theories that are deductively weak not only in respect to their arithmetical part but also in respect to their truth theoretic strength. Relative interpretability is sometimes understood as a relation of reduction. The interpretable theories of truth are deductively too weak to be interesting as an explication of a philosophical conception of truth besides a redundancy conception. CTT ∩ ITT contains only theories that are also weak in respect to their arithmetical part. Theories of ITT ∩ CTT , interpretable but nonconservative theories, are not as weak but they are quite artificial. Another reason is that Π1 -conservativity is also a measure of the arithmetical strength and therefore not directly connected to truth-theoretic strength. Interestingly all the theories of ITT are reflexive and not finitely axiomatizable and therefore similar to PA . Of more philosophical interest are the theories of CTT ∩ ITT , conservative extensions of PA that are not interpretable in PA . For deflationists conservativity over the base theory is a positive aspect of a theory of truth. It allows truth to be neutral and insubstantial. On the other hand some deflationists claim that the truth predicate fulfills an irreducible expressive function. So it would be an advantage if a theory of truth is deductively strong in respect to its truth-theoretic part. The noninterpretability of a theory in PA would be an indicator that the truththeoretic part of the theory cannot be ignored. The theories of CTT ∩ ITT are also of help in extracting the essentials of truth without influence of their arithmetical part. The set CTT ∩ ITT is important for deflationism, but not every theory that is an element of this set is as good as any other. A further investigation which gives more criteria would be of interest. None of the theories in − CTT ∩ ITT are reflexive and some of them like PT are − finitely axiomatizable. In this respect PT bears a resemblance to ACA 0 . There is more than just a resemblance, the two theories are equivalent in the − following sense: PT is a subtheory of a definitional extension of ACA 0 and the other way around. This can be − − seen as an argument for the ‘naturalness’ of PT . PT is also in other respects promising. Since it contains compositional axioms and a form of induction for formulas with the truth predicate the usual examples to show the deductive weakness of deflationist theories do not obtain. 97
It is an interesting open question if there are well motivated − truth-theoretic sentences not provable in PT . Literature Hájek, Petr and Pudlák, Pavel 1993 Metamathematics of First- Order Arithmetic, Berlin: Springer. Halbach, Volker 1999 “Conservative Theories of Classical Truth”, Studia Logica 62, 353-370. Kotlarski, Henryk, Krajewski, Stanislaw, and Lachlan, Alistair 1981 “Construction of Satisfaction Classes for Non-Standard Models”, Canadian Mathematical Bulletin 24, 283-293. Lindström, Per 1997 Aspects of Incompleteness, Berlin: Springer. Tarski, Alfred, Mostowski, Andrzej, and Robinson, Raphael M. 1953 Undecidable Theories, Amsterdam: North-Holland Pub. Co. 98 Interpretability Relations of Weak Theories of Truth — Martin Fischer
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31 Alexander Hieke Hannes Leitgeb H
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Reduktion und Elimination in Philos
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Distributors Die Österreichische L
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6 Inhalt / Contents Wittgenstein me
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8 Inhalt / Contents The Evolution o
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Formal Mechanisms for Reduction in
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4. Property language and theoretica
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imperatives, or commands in the for
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Referential Practice and the Lure o
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The Augustinian account confuses mo
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hurried to fix it on the page. The
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The Essence (?) of Color, According
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Literature Austin, James 1980 “Wi
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Wittgenstein’s Externalism - Gett
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Informal Reduction E.P. Brandon, Ca
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An Anti-Reductionist Argument Based
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An Anti-Reductionist Argument Based
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Did I Do It? -Yeah, You Did! Wittge
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Literature Did I Do It? -Yeah, You
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tween the two approaches. The task
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On Two Recent Defenses of The Simpl
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On Two Recent Defenses of The Simpl
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Queen Victoria’s Dying Thoughts T
Page 48 and 49: Diagonalization. The Liar Paradox,
Page 50 and 51: Proof: Given a function ƒ from con
Page 52 and 53: that I am sure that it is true that
Page 54 and 55: experiencing something as a red pen
Page 56 and 57: There can be Causal without Ontolog
Page 58 and 59: There can be Causal without Ontolog
Page 60 and 61: objects. But they also leave unansw
Page 62 and 63: The Knower Paradox and the Quantifi
Page 64 and 65: The Knower Paradox and the Quantifi
Page 66 and 67: lent. It also implies that an inven
Page 68 and 69: The Scapegoat Theory of Causality M
Page 70 and 71: We may have evolved a specialized c
Page 72 and 73: easoning have their source in the l
Page 74 and 75: Wittgenstein on Frazer and Explanat
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Page 78 and 79: an adequate syntactic analysis of o
Page 80 and 81: Wittgenstein meets ÖGS: Wovon man
Page 82 and 83: Wittgenstein meets ÖGS: Wovon man
Page 84 and 85: 3. Der Elementarsatz wird als Relat
Page 86 and 87: Literatur Glock, Hans Johann 2006
Page 88 and 89: Explaining the Brain: Ruthless Redu
Page 90 and 91: Occam’s Razor in the Theory of Th
Page 92 and 93: Literature Campbell, Donald T. 1987
Page 94 and 95: Die Nichtreduzierbarkeit der klassi
Page 96 and 97: Literatur Die Nichtreduzierbarkeit
Page 100 and 101: Does Bradley’s Regress Support No
Page 102 and 103: (3) Mutatis mutandis, realism has t
Page 104 and 105: Zeitliche Ontologie und zeitliche R
Page 106 and 107: Betrachtung ontologischer Fragen, M
Page 108 and 109: The decisive consequences of this a
Page 110 and 111: Benacerraf and Bad Company (An Atta
Page 112 and 113: offer a solution to the bad company
Page 114 and 115: Literature Benacerraf, Paul 1965
Page 116 and 117: "PA" by using "PA* ", at least at t
Page 118 and 119: Hard Naturalism and its Puzzles Ren
Page 120 and 121: The Mind-Body-Problem and Score-Kee
Page 122 and 123: mental properties, for instance, or
Page 124 and 125: kommt es zu einem gravierenden Miss
Page 126 and 127: Reduction Revisited: The Ontologica
Page 128 and 129: 4. Conclusion Reduction Revisited:
Page 130 and 131: All in all, we have seen that reduc
Page 132 and 133: Physicalism Without the A Priori Pa
Page 134 and 135: Literature Cartwright, Nancy 1999:
Page 136 and 137: Wittgensteins Projektionsmethode al
Page 138 and 139: Schlussbemerkungen Wittgensteins Pr
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Page 142 and 143: Relating Theories. Models and Struc
Page 144 and 145: Literature Relating Theories. Model
Page 146 and 147: e invoked in response to the questi
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Do Brains Think? Christopher Humphr
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4. Conclusion So do brains think or
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How Metaphors Alter the World-Pictu
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The Modal Supervenience of the Conc
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Literature van der Auwera, Johan an
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3. The Determination of Form by Syn
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Zwischen Humes Gesetz und „Sollen
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Assessing Humean Supervenience Amir
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determination thesis; it fails as a
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Aus der obigen Definition folgt, da
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Ding-Ontology of Aristotle vs. Sach
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Ding-Ontology of Aristotle vs. Sach
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How do Moral Principles Figure in M
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“Downward Causation”: Emergent,
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“Downward Causation”: Emergent,
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A with distinguished subset Z of te
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A Metaphysically Moderate Version o
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Literature Balashov, Yuri 2002 ''Wh
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“In der Frage liegt ein Fehler”
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Problems with Psychophysical Identi
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identity theory, or as a predecesso
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of the world. It only matters that
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Four Anti-reductionist Dogmas in th
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Four Anti-reductionist Dogmas in th
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notion of time and requires that so
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Some Remarks on Wittgenstein and Ty
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Some Remarks on Wittgenstein and Ty
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concrete particulars can be subject
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phenomenal concept of experiencing
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Metaphorische Bedeutung als virtus
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speaker is then attaching a special
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„Vom Weißdorn und vom Propheten
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„Die Einheit hören“ - Einige
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„Die Einheit hören“ - Einige
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S4) the necessity of arithmetic is
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Getting out from Inside: Why the Cl
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Dispensing with Particulars: Unders
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Literature Dispensing with Particul
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Reichenbach’s Concept of Logical
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Defining Ontological Naturalism Mar
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Literature Jackson, Frank 1986. “
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properties are located. This proces
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‘our conceptual scheme’ “is c
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Properties and Reduction between Me
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Does this mean that metaphysics sho
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do not pick out pain in a system by
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The Writing of Nietzsche and Wittge
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his teachings. He coincides with hi
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sense of a proposition without firs
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Naturalistic Ethics: A Logical Posi
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Literature Kant, Immanuel 2006 Crit
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What we now require is a bridge the
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Species, Variability, and Integrati
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could mention the many sorts of iso
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to work this way. Classic examples
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The Key Problems of KC Matteo Pleba
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have embraced the myth of prose: it
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5. Prior’s Problem As Patrick Bla
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Reductionism in Axiology: the Case
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developing of cancer tissue in huma
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(2) One-way Nagelian bridge princip
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Rethinking the Modal Argument again
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Rethinking the Modal Argument again
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that we are to find the features we
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Atypical Rational Agency Paul Raymo
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Literature Anscombe, G. E. M. 1957
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situation bieten sich keine Kriteri
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Two Reductions of ‘rule’ Dana R
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one that could take place, but does
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physics is not able to make, since
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Wittgenstein’s Attitudes Fabien S
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view of logic, in accordance with h
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Warum man auf transzendentalphiloso
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Making the Mind Higher-Level Elizab
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Literature Churchland, P. M. (1995)
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Mental Causation: A Lesson from Act
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Again, this is mistaken. Non-reduct
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auf alle Arten von kontingenten Pro
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Reduction, Sets, and Properties Ben
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Literature Egan, Andy (2004): ‘Se
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there is a tension between evaluati
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The Elimination of Meaning in Compu
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sufficient to explain all of the em
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that 'A' and 'A' could never be tok
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the necktie sense) can differ in wh
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Supervenience and ‘Should’ Arto
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word 'norm' can be thought to expre
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Rule-following as Coordination: A G
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vast majority are bent: gruesome, g
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human life, the violent condition o
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A Division in Mind. The Misconceive
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A Division in Mind. The Misconceive
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could then deduce (1c) and conclude
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Neutral Monism. A Miraculous, Incoh
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Neutral Monism. A Miraculous, Incoh
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A somewhat Eliminativist Proposal a
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Impliziert der intentionale Redukti
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Impliziert der intentionale Redukti
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Structure of the Paradoxes, Structu
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Structure of the Paradoxes, Structu
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that he was working on a translatio
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Objects of Perception, Objects of S
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Objects of Perception, Objects of S
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Let us take the example of the hypo
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Reducing Sets to Modalities Rafał
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variables ai that (under an interpr
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Are Lamarckian Explanations Fully R
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A Note on Tractatus 5.521 Nuno Vent
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And he goes on saying, referring to
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The Place of Theory Reduction in th
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Ethik als irreduzibles Supervenienz
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unabhängig von denen ihrer Mitglie
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Das ‘schwierige Problem’ des Be
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The Supervenience Argument, Levels,
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The Supervenience Argument, Levels,
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their relations both to the physica
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On the Characterization of Objects
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On the Characterization of Objects
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The Functional Unity of Special Sci
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size, location, temporal characteri
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Transcendental Philosophy and Mind-
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From Topology to Logic. The Neural
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From Topology to Logic. The Neural
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The Calculus of Inductive Construct
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The Four-Color Theorem, Testimony a
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experience. Of course, a detailed a
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Clearly x = y → x = ext y holds,
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Intentional Fundamentalism Petri Yl
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The second dimension to be consider
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The very existence of the special s
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Heil, J. (1999), “Multiple Realiz
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independent of what is the fact, fa
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415
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Publications of the Austrian Ludwig
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